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Shape and local growth for multidimensional branching random walks in random environment
Shape theorem subadditive ergodic theorem transience growth ex-ponent population size
2009/6/12
We study branching random walks in random environment on the d-dimensional square lattice, d 1. In this model, the environment has nite range dependence, and the population size cannot decrease. We pr...
A survey of results on random random walks on finite groups
random walk finite group Upper Bound Lemma Fourier analysis
2009/5/18
A number of papers have examined various aspects of ``random random walks'' on finite groups; the purpose of this article is to provide a survey of this work and to show, bring together, and discuss s...
Loop-Erased Random Walks, Spanning Trees and Hamiltonian Cycles
Loop-erased random walk spanning tree Wilson's algorithm
2009/5/4
We establish a formula for the distribution of loop-erased random walks at certain random times. Several classical results on spanning trees, including Wilson's algorithm, follow easily, as well as a ...
Pitman's 2M-X Theorem for Skip-Free Random Walks with Markovian Increments
Pitman's representation three-dimensional Bessel process telegrapher s equation queue Burke's theorem
2009/5/4
Let $(xi_k, kge 0)$ be a Markov chain on ${-1,+1}$ with $xi_0=1$ and transition probabilities $P(xi_{k+1}=1| xi_k=1)=a>b=P(xi_{k+1}=-1| xi_k=-1)$. Set $X_0=0$, $X_n=xi_1+cdots +xi_n$ and $M_n=max_{0le...
On Recurrent and Transient Sets of Inhomogeneous Symmetric Random Walks
Probabilities Wienertest Paley-Zygmund inequality
2009/5/4
We consider a continuous time random walk on the d-dimensional lattice Zd: the jump rates are time dependent, but symmetric and strongly elliptic with ellipticity constants independent of time. We inv...
A Non-Ballistic Law of Large Numbers for Random Walks in I.I.D. Random Environment
random walk in random environment RWRE law of large numbers
2009/4/29
We prove that random walks in i.i.d. random environments which oscillate in a given direction have velocity zero with respect to that direction. This complements existing results thus giving a general...
Geodesics and Recurrence of Random Walks in Disordered Systems
Random environment with stationary conductances Geodesicsin in first-passage percolation model Recurrence and transience
2009/4/29
In a first-passage percolation model on the square lattice $Z^2$, if the passage times are independent then the number of geodesics is either $0$ or $+infty$. If the passage times are stationary, ergo...
A Representation for Non-Colliding Random Walks
eigenvalues of random matrices Hermitian Brownian motion non-colliding Brownian motions queues in series Burke's theorem reversibility
2009/4/29
We define a sequence of mappings $Gamma_k:D_0(R_+)^kto D_0(R_+)^k$ and prove the following result: Let $N_1,ldots,N_n$ be the counting functions of independent Poisson processes on $R_+$ with respecti...
Recurrent Graphs where Two Independent Random Walks Collide Finitely Often
Random walk Comb lattice collisions
2009/4/28
We present a class of graphs where simple random walk is recurrent, yet two independent walkers meet only finitely many times almost surely. In particular, the comb lattice, obtained from $Z^2$ by rem...
A Log-scale Limit Theorem for One-dimensional Random Walks in Random Environments
RWRE limit theorems branching large deviations
2009/4/27
We consider a transient one-dimensional random walk X_n in random environment having zero asymptotic speed. For a class of non-i.i.d. environments we show that log X_n / log n converges in probability...
Recurrence and transience of excited random walks on $Z^d$ and strips
Excited Random Walk Recurrence Self-Interacting Random Walk Transience
2009/4/22
We investigate excited random walks on $Z^d, dge 1,$ and on planar strips $Ztimes{0,1,ldots,L-1}$ which have a drift in a given direction. The strength of the drift may depend on a random i.i.d. envir...
Random walks with k-wise independent increments
Random walk pseudo-randomness quasi-randomness pairwise independence
2009/4/22
We construct examples of a random walk with pairwise-independent steps which is almost surely bounded, and for any m and k a random walk with k-wise independent steps which has no stationary distribut...
Large Deviations for Local Times of Stable Processes and Stable Random Walks in 1 Dimension
Deviations Local Times Stable Processes Random Walks
2009/4/8
In Chen and Li (2004), large deviations were obtained for the spatial $L^p$ norms of products of independent Brownian local times and local times of random walks with finite second moment. The methods...
Recurrence and transience of excited random walks on $Z^d$ and strips
Recurrence transience random walk
2009/4/1
We investigate excited random walks on $Z^d, dge 1,$ and on planar strips $Ztimes{0,1,ldots,L-1}$ which have a drift in a given direction. The strength of the drift may depend on a random i.i.d. envir...
We construct examples of a random walk with pairwise-independent steps which is almost surely bounded, and for any m and k a random walk with k-wise independent steps which has no stationary distribut...